Non homogeneous quantum Markov states and quantum Markov fields

The program relative to the investigation of quantum Markov states for general one--dimensional spin models is carried on, following the strategy developed in the last years. In such a way, the emerging structure is fully clarified. This analysis is a starting point for the solution of the basic (still open) problem concerning the construction of a theory of quantum Markov fields, i.e. quantum Markov processes with multi--dimensional indices.